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Properties

Label 1-201-201.83-r1-0-0
Degree $1$
Conductor $201$
Sign $-0.946 - 0.323i$
Analytic cond. $21.6004$
Root an. cond. $21.6004$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.928 − 0.371i)2-s + (0.723 + 0.690i)4-s + (−0.841 + 0.540i)5-s + (−0.786 + 0.618i)7-s + (−0.415 − 0.909i)8-s + (0.981 − 0.189i)10-s + (0.888 + 0.458i)11-s + (0.580 + 0.814i)13-s + (0.959 − 0.281i)14-s + (0.0475 + 0.998i)16-s + (−0.723 + 0.690i)17-s + (−0.786 − 0.618i)19-s + (−0.981 − 0.189i)20-s + (−0.654 − 0.755i)22-s + (0.327 + 0.945i)23-s + ⋯
L(s)  = 1  + (−0.928 − 0.371i)2-s + (0.723 + 0.690i)4-s + (−0.841 + 0.540i)5-s + (−0.786 + 0.618i)7-s + (−0.415 − 0.909i)8-s + (0.981 − 0.189i)10-s + (0.888 + 0.458i)11-s + (0.580 + 0.814i)13-s + (0.959 − 0.281i)14-s + (0.0475 + 0.998i)16-s + (−0.723 + 0.690i)17-s + (−0.786 − 0.618i)19-s + (−0.981 − 0.189i)20-s + (−0.654 − 0.755i)22-s + (0.327 + 0.945i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.946 - 0.323i$
Analytic conductor: \(21.6004\)
Root analytic conductor: \(21.6004\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (1:\ ),\ -0.946 - 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02630552516 + 0.1584097440i\)
\(L(\frac12)\) \(\approx\) \(0.02630552516 + 0.1584097440i\)
\(L(1)\) \(\approx\) \(0.4956221785 + 0.09007305281i\)
\(L(1)\) \(\approx\) \(0.4956221785 + 0.09007305281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.928 - 0.371i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (-0.786 + 0.618i)T \)
11 \( 1 + (0.888 + 0.458i)T \)
13 \( 1 + (0.580 + 0.814i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (-0.786 - 0.618i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.235 - 0.971i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.981 - 0.189i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.723 - 0.690i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (-0.0475 - 0.998i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.38087383294334421650752200787, −25.0331435560989680209174133969, −24.62739058586790812881256693274, −23.24511686140128729196565107943, −22.82747393980313837198576183753, −21.01836096555292217489375031761, −19.97967212544689739890281353031, −19.55398307298962889994911545903, −18.53777014457236054269399367578, −17.2956634574251045748252541165, −16.45637450195769771198736931468, −15.84338970348024804635535860772, −14.77482909154927068229624696430, −13.43995743871763464324028243950, −12.193862897170261499353902967843, −11.117899302485954404998984978258, −10.158374663816554785333497097701, −8.914935639786574442129638160316, −8.21803644919903158052881273717, −6.98201053028174363140702273703, −6.11648655867113304250615461970, −4.46363481724744115981009774044, −3.11129233492517185953338282880, −1.10498629206690388027807005937, −0.08694340300027052964922790492, 1.78073820662699329407809026240, 3.14733374348197828155754409800, 4.149619089344989944611327673471, 6.45719909327656056639605067371, 6.93813018140917602107138145424, 8.4563481538327768872281251377, 9.15099254458407182001124393454, 10.35490496549881057917716829882, 11.44298587022936649113150746592, 12.07709847622979867545253925349, 13.26778168021617519563631414190, 15.01222798718036998348641082021, 15.62700693786858525300910371452, 16.6800653525324078691173328655, 17.70659024253944923013201041660, 18.86068837517068493612294540753, 19.35092680988988803652595169052, 20.11654063601359937126188849296, 21.541894212534579084649686410322, 22.208398061237784429038295045574, 23.38166452333500859590363150795, 24.52356329603196603252297913777, 25.81736409546557038610548615680, 26.04158906155596609545357020698, 27.35843900572235302983090416475

Graph of the $Z$-function along the critical line